betweenness               package:sna               R Documentation

_C_o_m_p_u_t_e _t_h_e _B_e_t_w_e_e_n_n_e_s_s _C_e_n_t_r_a_l_i_t_y _S_c_o_r_e_s _o_f _N_e_t_w_o_r_k _P_o_s_i_t_i_o_n_s

_D_e_s_c_r_i_p_t_i_o_n:

     `betweenness' takes a graph stack (`dat') and returns the
     betweenness centralities of positions within one graph (indicated
     by `nodes' and `g', respectively).  Depending on the specified
     mode, betweenness on directed or undirected geodesics will be
     returned; this function is compatible with `centralization', and
     will return the theoretical maximum absolute deviation (from
     maximum) conditional on size (which is used by `centralization' to
     normalize the observed centralization score).

_U_s_a_g_e:

     betweenness(dat, g=1, nodes=c(1:dim(dat)[2]), gmode="digraph", 
         diag=FALSE, tmaxdev=FALSE, cmode="directed", 
         geodist.precomp=NULL, rescale=FALSE)

_A_r_g_u_m_e_n_t_s:

     dat: Data array to be analyzed.  By assumption, the first
          dimension of the array indexes the graph, with the next two
          indexing the actors. Alternately, this can be an n x n matrix
          (if only one graph is involved). 

       g: Integer indicating the index of the graph for which
          centralities are to be calculated.  By default, `g'=1. 

   nodes: List indicating which nodes are to be included in the
          calculation.  By default, all nodes are included. 

   gmode: String indicating the type of graph being evaluated. 
          "digraph" indicates that edges should be interpreted as
          directed; "graph" indicates that edges are undirected. 
          `dmode' is set to "digraph" by default. 

    diag: Boolean indicating whether or not the diagonal should be
          treated as valid data.  Set this true if and only if the data
          can contain loops.  `diag' is `FALSE' by default. 

 tmaxdev: Boolean indicating whether or not the theoretical maximum
          absolute deviation from the maximum nodal centrality should
          be returned.  By default, `tmaxdev'==`FALSE'. 

   cmode: String indicating the type of betweenness centrality being
          computed (directed or undirected geodesics). 

geodist.precomp: A `geodist' object precomputed for the graph to be
          analyzed (optional) 

 rescale: If true, centrality scores are rescaled such that they sum to
          1. 

_D_e_t_a_i_l_s:

     The betweenness of a vertex, v, is given by


          C_B(v) = sum( g_ivj / g_ij, i,j: i!=j,i!=v,j!=v )


     where g_ijk is the number of geodesics from i to k through j. 
     Conceptually, high-betweenness vertices lie on a large number of
     non-redundant shortest paths between other vertices; they can thus
     be thought of as ``bridges'' or ``boundary spanners.''

_V_a_l_u_e:

     A vector containing the betweenness scores.

_W_a_r_n_i_n_g:

     Rescale may cause unexpected results if all actors have zero
     betweenness.

_N_o_t_e:

     Judicious use of `geodist.precomp' can save a great deal of time
     when computing multiple path-based indices on the same network.

_A_u_t_h_o_r(_s):

     Carter T. Butts ctb@andrew.cmu.edu

_R_e_f_e_r_e_n_c_e_s:

     Freeman, L.C.  (1979).  ``Centrality in Social Networks I:
     Conceptual Clarification.'' Social Networks, 1, 215-239.

_S_e_e _A_l_s_o:

     `centralization'

_E_x_a_m_p_l_e_s:

     g<-rgraph(10)     #Draw a random graph with 10 members
     betweenness(g)    #Compute betweenness scores

